Upper semicontinuity of the solution map to a parametric boundary optimal control problem with unbounded constraint sets

TL;DR

This paper investigates the stability of solutions in a parametric boundary optimal control problem governed by semilinear elliptic equations, focusing on upper semicontinuity under unbounded constraints and nonconvex costs.

Contribution

It provides sufficient conditions ensuring the upper semicontinuity and continuity of the solution map in complex control problems with unbounded admissible sets.

Findings

Established conditions for upper semicontinuity of the solution map.

Proved continuity of the solution map under certain parameter variations.

Addressed stability issues in nonconvex, unbounded control problems.

Abstract

We would like to study the solution stability of a parametric control problem governed by semilinear elliptic equations with a mixed state-control constraint, where the cost function is nonconvex and the admissible set is unbounded. The main goal of this paper is to give some sufficient conditions under which the solution map is upper semicontinuous and continuous with respect to parameters.